The origins of this plant are the subject of controversy, with most authorities claiming it is native to South Asia (particularly the Ganges Delta), while others claim its origin is in northwestern South America.

Why origins is used in plural at the beginning and later switched to singular? Can I say "a subject of controversy" instead of "the subject of controversy?

Thanks in advance!
A very subtle distinction. In this context, I suspect because the first part of the sentence is suggesting that there are several theories, whereas the second use is a specific placing.
You could use "a subject of controversy".
The first 'origins' refers to any of several possible sources for the plant (and some hybrids can originate in more than one place), while the second 'origin' refers to one specific possibility-- NW South America.

Yes, you could use the indefinite article here.